Arithmetic Billiards

Outer Billiards, Arithmetic Graphs and Multigrid Flows

I considered the first construction in my book, Outer Billiards on Kites [S2]. The arithmetic graphs served as the main tool for understanding the dynamics of outer billiards on kites. See §2 for definitions. The second construction, which is quite general, is based on patterns of oriented lines in the Euclidean plane. See §3 for definitions. The concrete instance of the multigrid construction ...

Track Billiards

We study a class of planar billiards having the remarkable property that their phase space consists up to a set of zero measure of two invariant sets formed by orbits moving in opposite directions. The tables of these billiards are tubular neighborhoods of differentiable Jordan curves that are unions of finitely many segments and arcs of circles. We prove that under proper conditions on the seg...

Quantum Billiards

Rigid-body motion, interacting billiards, and billiards on curved manifolds.

It is shown that the free motion of any three-dimensional rigid body colliding elastically between two parallel, at walls is equivalent to a three-dimensional billiard system. Depending upon the inertial parameters of the problem, the billiard system may possess a potential energy eld and a non-Euclidean connguration space. The corresponding curvilinear motion of the billiard ball does not nece...

ARITHMETIC-BASED FUZZY CONTROL

Fuzzy control is one of the most important parts of fuzzy theory for which several approaches exist. Mamdani uses $alpha$-cuts and builds the union of the membership functions which is called the aggregated consequence function. The resulting function is the starting point of the defuzzification process. In this article, we define a more natural way to calculate the aggregated consequence funct...